One 'real world calculation' that you might do is take the derivative of that function in some problem that you might encounter if you are an engineer or scientist or something like that. The concept of ##\sin{x}## function and the ##x## function themselves lead to more real world applications than you could ever count, but the quotient ##\sin{x}/x## seems to be more interesting when viewed from a mathematical perspective.

The sinc function: ##\mathrm{sinc}(x) \equiv \dfrac{\sin x}{x}## is the Fourier transform of a rectangular window or pulse. As a result, there are many real-world applications to signal processing. In particular, ##\mathrm{sinc}(x)## is the kernel for continuous reconstruction of a bandlimited, discrete-sampled signal (e.g., digital audio).

That sort of question might be a sign of taking the idea of "real-world" applicability a little too far. Sometimes, things can have a real-world "use" if they merely strengthen your understanding of real world phenomenon, rather than being directly applicable. It can sometimes be a piece of the fabric that holds a subject together.

In contrast, I got disillusioned with topology on the basis of entire books and hundreds and hundreds of pages that I could not see any real world application for, as opposed to one single function that can fit in one line, which might serve as a useful example to illustrate the theory. That should put it into perspective. I don't think it make sense to question the applicability of every little example. It's just something to practice on and gain experience with the subject, and that is real-world applicability. I'm not saying it's not a fair question to ask or an interesting question, but the way it is phrased suggests there might be some kind of underlying over-thinking going on.